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Example text
For p = 1, the proof is straightforward. ✷ We are now in a position to prove boundedness from L1 into L∞ . 2. For every t > 0 we have sup P α (t; x, y) < +∞. (16) x,y∈R2 Proof. To achieve this goal, we are going to use the construction of the Hamiltonian Hα via Dirichlet forms as done in [3] which we recall here. We omit the case α = ∞, which corresponds to the free Hamiltonian. Let ϕα be the following function ϕα (x) = H0(1)(2ie−2πα+ (1) |x|), x ∈ R2 \ {0}, (17) where H0(1) is the Hankel function.
Voigt and P. Stollmann [15, 16] and the new functional calculus developed by E. B. Davies [8], we prove the pindependence of their spectra and derive the exponential growth of the p-norm of exp(−tHα,p ). Such a construction was done by S. Albeverio et al. [2] using the ‘family of pseudo-resolvent’, thereby exploiting the expression of the resolvent kernel of Hα . They conclude the construction for d = 1, p ∈ [1, +∞) or d = 2, p ∈ ]1, +∞[ or d = 3, p ∈ ] 32 , 3[ and the same thing for the C0 -semigroup exp(−tHα ).
Geom. 7 (2000), 266–283 (Russian). Voiculescu, D. : Limit laws for random matrices and free products, Invent. Math. 104 (1991), 201–220. : A strengthened asymptotic freeness result for random matrices with applications to free entropy, Internat. Math. Res. Notices 1998, No. 1, 41–62. Voiculescu, D. , Dykema, K. : Free Random Variables, CRM Monograph Series, Amer. Math. , Providence, RI, 1992. Mathematical Physics, Analysis and Geometry 4: 37–49, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.